Physics 207 Lecture 13. Lecture 13

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1 Physcs 07 Lecture 3 Goals: Lecture 3 Chapter 0 Understand the relatonshp between moton and energy Defne Potental Energy n a Hooke s Law sprng Develop and explot conservaton of energy prncple n problem solvng Chapter Understand the relatonshp between force, dsplacement and work Assgnment: HW6 due Wednesday, Feb. For Thursday: Read all of Chapter Physcs 07: Lecture 3, Pg Energy -mg y= ½ m (v y - v y0 ) -mg (y f y ) = ½ m ( v yf -v y ) A relatonshp between y-dsplacement and change n the y-speed Rearrangng to gve ntal on the left and fnal on the rght ½ m v y + mgy = ½ m v yf + mgy f We now defne mgy as the gravtatonal potental energy Physcs 07: Lecture 3, Pg Page

2 Physcs 07 Lecture 3 Energy Notce that f we only consder gravty as the external force then the x and z veloctes reman constant To ½ m v y + mgy = ½ m v yf + mgy f Add ½ m v x + ½ m v z and ½ m v xf + ½ m v zf ½ m v + mgy = ½ m v f + mgy f where v = v x +v y + v z ½ m v terms are defned to be knetc energes (A scalar quantty of moton) Physcs 07: Lecture 3, Pg 3 Energy If only conservatve forces are present, the total energy (sum of potental, U, and knetc energes, K) ) of a system s conserved For an object n a gravtatonal feld ½ m v y + mgy = ½ m v yf + mgy f K ½ mv U mgy E mech = K + U E mech = K + U = constant K and U may change, but E mech = K + U remans a fxed value. E mech s called mechancal energy Physcs 07: Lecture 3, Pg 4 Page

3 Physcs 07 Lecture 3 Example of a conservatve system: The smple pendulum. Suppose we release a mass m from rest a dstance h above ts lowest possble pont. What s the maxmum speed of the mass and where does ths happen? To what heght h does t rse on the other sde? m h h v Physcs 07: Lecture 3, Pg 5 Example: The smple pendulum. What s the maxmum speed of the mass and where does ths happen? E = K + U = constant and so K s maxmum when U s a mnmum. y y=h y=0 Physcs 07: Lecture 3, Pg 6 Page 3

4 Physcs 07 Lecture 3 Example: The smple pendulum. What s the maxmum speed of the mass and where does ths happen? E = K + U = constant and so K s maxmum when U s a mnmum E = mgh at top E = mgh = ½ mv at bottom of the swng y y=h y=0 h v Physcs 07: Lecture 3, Pg 7 Example: The smple pendulum. To what heght h does t rse on the other sde? E = K + U = constant and so when U s maxmum agan (when K = 0) t wll be at ts hghest pont. E = mgh = mgh or h = h y y=h =h y=0 Physcs 07: Lecture 3, Pg 8 Page 4

5 Physcs 07 Lecture 3 Example The Loop-the-Loop agan To complete the loop the loop, how hgh do we have to let the release the car? Condton for completng the loop the loop: Crcular moton at the top of the loop (a c = v / R) Use fact that E = U + K = constant! y=0 U b =mgh Recall that g s the source of Car has mass m U=mgR the centrpetal acceleraton and N just goes to zero s the lmtng case. h? Also recall the mnmum R speed at the top s v = (A) R (B) 3R (C) 5/ R (D) 3/ R Physcs 07: Lecture 3, Pg 9 gr Example The Loop-the-Loop agan Use E = K + U = constant mgh + 0 = mg R + ½ mv mgh = mg R + ½ mgr = 5/ mgr v = gr h = 5/ R h? R Physcs 07: Lecture 3, Pg 0 Page 5

6 Physcs 07 Lecture 3 What speed wll the skateboarder reach halfway down the hll f there s no frcton and the skateboarder starts at rest? Assume we can treat the skateboarder as a pont Assume zero of gravtatonal U s at bottom of the hll R=0 m m = 5 kg 30 R=0 m Example Skateboard y=0 Physcs 07: Lecture 3, Pg What speed wll the skateboarder reach halfway down the hll f there s no frcton and the skateboarder starts at rest? Assume we can treat the skateboarder as pont Assume zero of gravtatonal U s at bottom of the hll R=0 m m = 5 kg 30 R=0 m Example Skateboard Use E = K + U = constant E before = E after 0 + m g R = ½ mv + mgr (-sn 30 ) mgr/ = ½ mv gr = v v= (gr) ½ v = (0 x 0) ½ = 0 m/s Physcs 07: Lecture 3, Pg Page 6

7 Physcs 07 Lecture 3 Potental Energy, Energy Transfer and Path A ball of mass m, ntally at rest, s released and follows three dfference paths. All surfaces are frctonless. The ball s dropped. The ball sldes down a straght nclne 3. The ball sldes down a curved nclne After travelng a vertcal dstance h, how do the three speeds compare? 3 h (A) > > 3 (B) 3 > > (C) 3 = = (D) Can t tell Physcs 07: Lecture 3, Pg 3 Potental Energy, Energy Transfer and Path A ball of mass m, ntally at rest, s released and follows three dfference paths. All surfaces are frctonless. The ball s dropped. The ball sldes down a straght nclne 3. The ball sldes down a curved nclne After travelng a vertcal dstance h, how do the three speeds compare? 3 h (A) > > 3 (B) 3 > > (C) 3 = = (D) Can t tell Physcs 07: Lecture 3, Pg 4 Page 7

8 Physcs 07 Lecture 3 Example Skateboard What s the normal force on the skate boarder? N R=0 m m = 5 kg mg R=0 m Physcs 07: Lecture 3, Pg 5 Example Skateboard Now what s the normal force on the skate boarder? N R=0 m m = 5 kg 30 R=0 m 60 mg Σ F r = ma r = m v / R = N mg cos 60 N = m v /R + mg cos 60 N = 5 00 / (0.87) N = =470 Newtons Physcs 07: Lecture 3, Pg 6 Page 8

9 Physcs 07 Lecture 3 Elastc vs. Inelastc Collsons A collson s sad to be elastc when energy as well as momentum s conserved before and after the collson. K before = K after Carts colldng wth a perfect sprng, bllard balls, etc. v Physcs 07: Lecture 3, Pg 7 Elastc vs. Inelastc Collsons A collson s sad to be nelastc when energy s not conserved before and after the collson, but momentum s conserved. K before K after Car crashes, collsons where objects stck together, etc. Physcs 07: Lecture 3, Pg 8 Page 9

10 Physcs 07 Lecture 3 Inelastc collson n -D: Example A block of mass M s ntally at rest on a frctonless horzontal surface. A bullet of mass m s fred at the block wth a muzzle velocty (speed) v. The bullet lodges n the block, and the block ends up wth a speed V. What s the ntal energy of the system? What s the fnal energy of the system? Is energy conserved? v V x before after Physcs 07: Lecture 3, Pg 9 Inelastc collson n -D: Example What s the momentum of the bullet wth speed v? What s the ntal energy of the system? mv r m v r v r = mv What s the fnal energy of the system? ( m + M )V Is momentum conserved (yes)? m v + M 0 = ( m + M )V Is energy conserved? Examne E before -E after mv [( m + M )V]V = mv m ( mv) m + M m v = mv ( m + M ) v No! V before after x Physcs 07: Lecture 3, Pg 0 Page 0

11 Physcs 07 Lecture 3 Example Fully Elastc Collson Suppose I have dentcal bumper cars. One s motonless and the other s approachng t wth velocty v. If they collde elastcally, what s the fnal velocty of each car? Identcal means m = m = m Intally v Green = v and v Red = 0 COM mv + 0 = mv f + mv f v = v f + v f COE ½ mv = ½ mv f + ½ mv f v = v f + v f v = (v f + v f ) = v f +v f v f + v f v f v f = 0 Soln : v f = 0 and v f = v Soln : v f = 0 and v f = v Physcs 07: Lecture 3, Pg Varable force devces: Hooke s Law Sprngs Sprngs are everywhere, (probe mcroscopes, DNA, an effectve nteracton between atoms) F In ths sprng, the magntude of the force ncreases as the sprng s further compressed (a dsplacement). Hooke s Law, F s = - k s s Rest or equlbrum poston s s the amount the sprng s stretched or compressed from t restng poston. Physcs 07: Lecture 3, Pg Page

12 Physcs 07 Lecture 3 Exercse Hooke s Law 8 m 9 m What s the sprng constant k? 50 kg (A) 50 N/m (B) 00 N/m (C) 400 N/m (D) 500 N/m Physcs 07: Lecture 3, Pg 3 Exercse Hooke s Law 8 m 9 m What s the sprng constant k? 50 kg F sprng ΣF = 0 = F s mg = k s - mg Use k = mg/ s = 500 N /.0 m (A) 50 N/m (B) 00 N/m (C) 400 N/m (D) 500 N/m mg Physcs 07: Lecture 3, Pg 4 Page

13 Physcs 07 Lecture 3 F-s relaton for a foot arch: Force (N) Dsplacement (mm) Physcs 07: Lecture 3, Pg 5 Force vs. Energy for a Hooke s Law sprng F = - k (x x equlbrum ) F = ma = m dv/dt = m (dv/dx dx/dt) = m dv/dx v = mv dv/dx So - k (x x equlbrum ) dx = mv dv Let u = x x eq. & du = dx f xf vf m ku du= mv dv x kx + kx = v x f ku x = mv mv f v v f mv kx + mv = kx + f mv f Physcs 07: Lecture 3, Pg 6 Page 3

14 Physcs 07 Lecture 3 Energy for a Hooke s Law sprng kx + mv = kx + f mv f Assocate ½ kx wth the potental energy of the sprng m U s + K = U + sf K f Hooke s Law sprngs are conservatve so the mechancal energy s constant Physcs 07: Lecture 3, Pg 7 In general: Energy dagrams Energy Ball fallng E mech K U Energy Sprng/Mass system E mech K U y s Physcs 07: Lecture 3, Pg 8 Page 4

15 Physcs 07 Lecture 3 Energy dagrams Sprng/Mass/Gravty system Force sprng alone y -mg sprng & gravty m Energy E mech K K U g U sprng U Total y Physcs 07: Lecture 3, Pg 9 Equlbrum Example Sprng: F x = 0 => du / dx = 0 for x=x eq The sprng s n equlbrum poston In general: du / dx = 0 equlbrum for ANY functon establshes U U stable equlbrum unstable equlbrum Physcs 07: Lecture 3, Pg 30 Page 5

16 Physcs 07 Lecture 3 Comment on Energy Conservaton We have seen that the total knetc energy of a system undergong an nelastc collson s not conserved. Mechancal energy s lost: Heat (frcton) Bendng of metal and deformaton Knetc energy s not conserved by these non-conservatve forces occurrng durng the collson! Momentum along a specfc drecton s conserved when there are no external forces actng n ths drecton. In general, easer to satsfy conservaton of momentum than energy conservaton. Physcs 07: Lecture 3, Pg 3 Lecture 3 Assgnment: HW6 due Wednesday / For Monday: Read all of chapter Physcs 07: Lecture 3, Pg 3 Page 6